// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012 Giacomo Po <gpo@ucla.edu>
// Copyright (C) 2011-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2018 David Hyde <dabh@stanford.edu>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_MINRES_H_
#define EIGEN_MINRES_H_

namespace Eigen {

namespace internal {

    /** \internal Low-level MINRES algorithm
         * \param mat The matrix A
         * \param rhs The right hand side vector b
         * \param x On input and initial solution, on output the computed solution.
         * \param precond A right preconditioner being able to efficiently solve for an
         *                approximation of Ax=b (regardless of b)
         * \param iters On input the max number of iteration, on output the number of performed iterations.
         * \param tol_error On input the tolerance error, on output an estimation of the relative error.
         */
    template <typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
    EIGEN_DONT_INLINE void
    minres(const MatrixType& mat, const Rhs& rhs, Dest& x, const Preconditioner& precond, Index& iters, typename Dest::RealScalar& tol_error)
    {
        using std::sqrt;
        typedef typename Dest::RealScalar RealScalar;
        typedef typename Dest::Scalar Scalar;
        typedef Matrix<Scalar, Dynamic, 1> VectorType;

        // Check for zero rhs
        const RealScalar rhsNorm2(rhs.squaredNorm());
        if (rhsNorm2 == 0)
        {
            x.setZero();
            iters = 0;
            tol_error = 0;
            return;
        }

        // initialize
        const Index maxIters(iters);                                    // initialize maxIters to iters
        const Index N(mat.cols());                                      // the size of the matrix
        const RealScalar threshold2(tol_error * tol_error * rhsNorm2);  // convergence threshold (compared to residualNorm2)

        // Initialize preconditioned Lanczos
        VectorType v_old(N);                // will be initialized inside loop
        VectorType v(VectorType::Zero(N));  //initialize v
        VectorType v_new(rhs - mat * x);    //initialize v_new
        RealScalar residualNorm2(v_new.squaredNorm());
        VectorType w(N);                         // will be initialized inside loop
        VectorType w_new(precond.solve(v_new));  // initialize w_new
                                                 //            RealScalar beta; // will be initialized inside loop
        RealScalar beta_new2(v_new.dot(w_new));
        eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
        RealScalar beta_new(sqrt(beta_new2));
        const RealScalar beta_one(beta_new);
        // Initialize other variables
        RealScalar c(1.0);  // the cosine of the Givens rotation
        RealScalar c_old(1.0);
        RealScalar s(0.0);                      // the sine of the Givens rotation
        RealScalar s_old(0.0);                  // the sine of the Givens rotation
        VectorType p_oold(N);                   // will be initialized in loop
        VectorType p_old(VectorType::Zero(N));  // initialize p_old=0
        VectorType p(p_old);                    // initialize p=0
        RealScalar eta(1.0);

        iters = 0;  // reset iters
        while (iters < maxIters)
        {
            // Preconditioned Lanczos
            /* Note that there are 4 variants on the Lanczos algorithm. These are
                 * described in Paige, C. C. (1972). Computational variants of
                 * the Lanczos method for the eigenproblem. IMA Journal of Applied
                 * Mathematics, 10(3), 373-381. The current implementation corresponds 
                 * to the case A(2,7) in the paper. It also corresponds to 
                 * algorithm 6.14 in Y. Saad, Iterative Methods for Sparse Linear
                 * Systems, 2003 p.173. For the preconditioned version see 
                 * A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM (1987).
                 */
            const RealScalar beta(beta_new);
            v_old = v;                                 // update: at first time step, this makes v_old = 0 so value of beta doesn't matter
            v_new /= beta_new;                         // overwrite v_new for next iteration
            w_new /= beta_new;                         // overwrite w_new for next iteration
            v = v_new;                                 // update
            w = w_new;                                 // update
            v_new.noalias() = mat * w - beta * v_old;  // compute v_new
            const RealScalar alpha = v_new.dot(w);
            v_new -= alpha * v;            // overwrite v_new
            w_new = precond.solve(v_new);  // overwrite w_new
            beta_new2 = v_new.dot(w_new);  // compute beta_new
            eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
            beta_new = sqrt(beta_new2);  // compute beta_new

            // Givens rotation
            const RealScalar r2 = s * alpha + c * c_old * beta;  // s, s_old, c and c_old are still from previous iteration
            const RealScalar r3 = s_old * beta;                  // s, s_old, c and c_old are still from previous iteration
            const RealScalar r1_hat = c * alpha - c_old * s * beta;
            const RealScalar r1 = sqrt(std::pow(r1_hat, 2) + std::pow(beta_new, 2));
            c_old = c;          // store for next iteration
            s_old = s;          // store for next iteration
            c = r1_hat / r1;    // new cosine
            s = beta_new / r1;  // new sine

            // Update solution
            p_oold = p_old;
            p_old = p;
            p.noalias() = (w - r2 * p_old - r3 * p_oold) / r1;  // IS NOALIAS REQUIRED?
            x += beta_one * c * eta * p;

            /* Update the squared residual. Note that this is the estimated residual.
                The real residual |Ax-b|^2 may be slightly larger */
            residualNorm2 *= s * s;

            if (residualNorm2 < threshold2)
            {
                break;
            }

            eta = -s * eta;  // update eta
            iters++;         // increment iteration number (for output purposes)
        }

        /* Compute error. Note that this is the estimated error. The real 
             error |Ax-b|/|b| may be slightly larger */
        tol_error = std::sqrt(residualNorm2 / rhsNorm2);
    }

}  // namespace internal

template <typename _MatrixType, int _UpLo = Lower, typename _Preconditioner = IdentityPreconditioner> class MINRES;

namespace internal {

    template <typename _MatrixType, int _UpLo, typename _Preconditioner> struct traits<MINRES<_MatrixType, _UpLo, _Preconditioner>>
    {
        typedef _MatrixType MatrixType;
        typedef _Preconditioner Preconditioner;
    };

}  // namespace internal

/** \ingroup IterativeLinearSolvers_Module
     * \brief A minimal residual solver for sparse symmetric problems
     *
     * This class allows to solve for A.x = b sparse linear problems using the MINRES algorithm
     * of Paige and Saunders (1975). The sparse matrix A must be symmetric (possibly indefinite).
     * The vectors x and b can be either dense or sparse.
     *
     * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
     * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower,
     *               Upper, or Lower|Upper in which the full matrix entries will be considered. Default is Lower.
     * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
     *
     * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
     * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
     * and NumTraits<Scalar>::epsilon() for the tolerance.
     *
     * This class can be used as the direct solver classes. Here is a typical usage example:
     * \code
     * int n = 10000;
     * VectorXd x(n), b(n);
     * SparseMatrix<double> A(n,n);
     * // fill A and b
     * MINRES<SparseMatrix<double> > mr;
     * mr.compute(A);
     * x = mr.solve(b);
     * std::cout << "#iterations:     " << mr.iterations() << std::endl;
     * std::cout << "estimated error: " << mr.error()      << std::endl;
     * // update b, and solve again
     * x = mr.solve(b);
     * \endcode
     *
     * By default the iterations start with x=0 as an initial guess of the solution.
     * One can control the start using the solveWithGuess() method.
     *
     * MINRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
     *
     * \sa class ConjugateGradient, BiCGSTAB, SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
     */
template <typename _MatrixType, int _UpLo, typename _Preconditioner> class MINRES : public IterativeSolverBase<MINRES<_MatrixType, _UpLo, _Preconditioner>>
{
    typedef IterativeSolverBase<MINRES> Base;
    using Base::m_error;
    using Base::m_info;
    using Base::m_isInitialized;
    using Base::m_iterations;
    using Base::matrix;

public:
    using Base::_solve_impl;
    typedef _MatrixType MatrixType;
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef _Preconditioner Preconditioner;

    enum
    {
        UpLo = _UpLo
    };

public:
    /** Default constructor. */
    MINRES() : Base() {}

    /** Initialize the solver with matrix \a A for further \c Ax=b solving.
         *
         * This constructor is a shortcut for the default constructor followed
         * by a call to compute().
         *
         * \warning this class stores a reference to the matrix A as well as some
         * precomputed values that depend on it. Therefore, if \a A is changed
         * this class becomes invalid. Call compute() to update it with the new
         * matrix A, or modify a copy of A.
         */
    template <typename MatrixDerived> explicit MINRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}

    /** Destructor. */
    ~MINRES() {}

    /** \internal */
    template <typename Rhs, typename Dest> void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
    {
        typedef typename Base::MatrixWrapper MatrixWrapper;
        typedef typename Base::ActualMatrixType ActualMatrixType;
        enum
        {
            TransposeInput = (!MatrixWrapper::MatrixFree) && (UpLo == (Lower | Upper)) && (!MatrixType::IsRowMajor) && (!NumTraits<Scalar>::IsComplex)
        };
        typedef typename internal::conditional<TransposeInput, Transpose<const ActualMatrixType>, ActualMatrixType const&>::type RowMajorWrapper;
        EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(MatrixWrapper::MatrixFree, UpLo == (Lower | Upper)),
                            MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY);
        typedef typename internal::conditional<UpLo == (Lower | Upper),
                                               RowMajorWrapper,
                                               typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::Type>::type SelfAdjointWrapper;

        m_iterations = Base::maxIterations();
        m_error = Base::m_tolerance;
        RowMajorWrapper row_mat(matrix());
        internal::minres(SelfAdjointWrapper(row_mat), b, x, Base::m_preconditioner, m_iterations, m_error);
        m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
    }

protected:
};

}  // end namespace Eigen

#endif  // EIGEN_MINRES_H
